Elementary 
Industrial  School 


Cleveland  Public  School* 


MATHEMATICS 


The  Board  of  Education 
1912 


Elementary 
Industrial  School 

Cleveland  Public  Schools 


MATHEMATICS 


The  Board  of  Education 
1912 


The  work  in  mathematics  herein  outlined  is  based  upon  three 
years'  experience  with  classes  in  the  Cleveland  Elementary  Industrial 
School. 

Much  of  the  work  usually  taught  in  higher  grammar  grades  is 
touched  upon  lightly,  or  omitted  altogether.  Nothing  is  omitted,  how- 
ever, which  has  apparent  practical  value  in  the  life  of  the  pupil,  and 
nothing  that  is  essential  to  the  study  of  the  higher  mathematics. 

The  value  of  abstract  work  in  dealing  with  principles  is  conceded, 
but  the  aim  is  to  make  the  work  of  practical  value  as  a  part  of  the  daily 
life  experience  of  the  pupil.  To  this  end  concrete  examples  are  par- 
ticularly emphasized,  and  all  possible  material  is  drawn  from  the  prob- 
lems which  the  pupil  meets  in  his  occupational  or  other  school  work 
and  in  connection  with  his  home  and  community  life.  To  the  teacher 
correlation  thus  becomes  a  vitally  important  element  in  class  work. 
To  the  pupil  knowledge  becomes  desirable  because  immediately  neces- 
sary to  success. 

The  work  covers  two  years  corresponding  to  the  seventh  and 
eighth  grades  of  the  elementary  schools.  The  method  used  is  partly 
Topical  and  partly  Spiral.  The  correlation  work  demands  the  Spiral 
method,  especially  during  the  first  year,  and  it  is  advisable  because 
pupils  are  frequently  entering  the  classes  from  the  regular  elementary 
schools. 

The  outline  is  the  work  of  Miss  Myrtle  G.  Campbell,  instructor  of 
mathematics  in  the  Elementary  Industrial  School. 

HARRIET  L.  KEELER, 

Superintendent  of  Schools. 
August,  1912, 


CONTENTS 

I     Integers     (a)  Roman  Numerals  and  Arabic  Characters 

(b)  Arabic  Notation  and  Numeration 

(c)  Fundamental  Processes  and  Principles 

(d)  Powers  and  Factors 

(e)  Classification  of  Numbers 
II     Fractions  (a)  Common  Fractions. 

(b)     Decimal  Fractions.  , 

III  Compound  Numbers 

IV  United  States  Money 

V  Percentage  and  its  Applications 

VI  Ratio 

VII  Proportion  and  The  Equation 

VIII  Mensuration  and  Construction  Problems 

IX  General  Suggestions 

OUTLINE  FOR  TWO  YEARS'  WORK 

I     INTEGERS 

(a)  Roman  Numerals  and  Arabic  Characters 

Brief  History 

Reading  and  writing  of  Roman  Numerals  from  one  to  one 
hundred,  and  by  hundreds  to  one  thousand. 

Suggestions  for  Application 

Reading  and  writing  of  chapter  numbers,  dates,  stone  cuttings, 
etc.;  enumerating;  classifying. 

(b)  Arabic  Notation  and  Numeration 

Use  of  numbers  through  billions. 

Suggestions  for  Application 

Writing  numbers  as  used  on  bank  checks,  etc. 

Reading  numbers  which  express  commonly  used  astronomical 
distances.  £frfftyT^liij 

Suggestions  for  Correlation 

Making  of  graphs  showing  products  of  the  great  industries, 
effects  of  inventions,  etc. 

Reading  of  official  reports  and  newspaper  articles,  giving 
populations,  municipal  and  national  government  expenses,  year's 
agricultural  crops,  etc. 

(c)  Fundamental  Processes  and  Principles 

Horizontal  addition  of  numbers  less  than  one  hundred. 
Finding  missing  terms  in  multiplication  as  a  foundation  for 
the  subject  of  Percentage. 

Emphasizing  practical  short  methods  including  cancellation. 
Importance  attached  to  checking. 


W65687 


Suggestions  for  Application 

Marketing  and  shop-keeping  in  the  class-room,  emphasizing 
the  following  subjects:  Study  of  Prices;  Making  Change  (using 
college  currency)  ;  Need  and  Use  of  Sales  Checks  and  Cash- 
books;  Meaning  of  C.  O.  D.,  Charge,  Debit,  and  Credit;  Bills: 
Written,  Discounted  and  Receipted. 

Suggestions  for  Correlation 

Use  actual  experiences  received  in  the  occupational  work  of 
the  school  in  making  payrolls,  noting  the  time  required. 
Give  emphasis  to  accuracy  and  neatness. 
Use  of  Piece  Labor  Cards. 

Stimulate  just  social  relations  between  employer  and  em- 
ployee. 

(d)  Powers  and  Factors 

Review  of  the  Multiplication  Table  and  the  powers  and 
multiples  selected. 

Squares  of  numbers  one  to  twenty-five  memorized,  and  num- 
bers twenty-five  to  one  hundred  squared  mentally  by  algebraic 
formula. 

Least  Common  Multiple  and  finding  common  denominators 
of  fractions  commonly  used  in  correlation  and  in  business. 

Tests  for  factors. 

Greatest  Common  Divisor. 

Square  Root  as  needed  for  mensuration  and  correlation 
problems. 

Square  Root  by  factoring  method. 

The  square  root  of  two  and  the  square  root  of  three  mem- 
orized. 

Cubes  memorized  from  one  to  twelve,  inclusive. 

Suggestions  for  Application 

Least  Common  Multiple  as  a  common  container  compared 
with  capacity  of  crates,  packing  boxes,  cases,  bins,  etc. 

Width  of  wall-paper  may  be  a  common  divisor  for  walls. 

(e)  Classification  of  Numbers 

Meaning  and  use  of  the  class  names:  Integral;  Fractional; 
Mixed ;  Odd  and  Even ;  Prime  and  Composite ;  Abstract  and  Con- 
crete. 

II     FRACTIONS 
(a)     Common  Fractions 

Reductions  and  Principles. 
Fundamental  Processes. 

Special  emphasis  put  upon  the  three  types  of  problems  which 
make  the  foundation  for  Percentage. 

Practical  Aliquot  Parts,  especially  sixths,  eighths,  and 
twelfths  of  one  hundred. 


Suggestions  for  Application 

Drill  in  finding  the  price  of  one  unit  of  measure  when  bought 
and  sold,  three,  six,  eight,  or  one  dozen  for  one  dollar. 

Use  short  method  for  multiplication  when  aliquot  parts  are 
given  as  prices. 

Find  pitch  of  roofs,  expressing  heights  as  numerators  and 
widths  of  houses  as  denominators. 

Suggestions  for  Correlation 

Give  original  problems  suggested  in  shop  work,  using  one- 
eighth,  one-fourth  and  three-eighths  of  an  inch  for  partitions  and 
sides  of  boxes,  trays,  etc. 

Allow  one-fourth  inch  for  finish  and  one-eighth  inch  for  saw- 
curfs,  in  cutting  material  for  class  supplies. 

Allowing  for  finish,  find  the  total  amount  of  lumber  required 
for  pieces  of  furniture  when  mill  bills  are  given. 

In  sewing  allow  for  waste  in  cutting  out  garments,  and  esti- 
mate the  number  of  yards,  pieces  or  bolts  of  material  needed  for 
supplying  classes. 

(b)     Decimal  Fractions 

Compared  with  Common  Fractions. 

Notation  and  Numeration. 

Reduction  and  Principles. 

Fundamental  Processes  using  three  decimal  places. 

Suggestions  for  Application 

•Reading  and  writing  numbers  which  express  the  velocity  of 
light  waves  and  sound  waves. 

Specific  gravity. 

Practical  use  of  Average  in  finding  the  average  depth,  speed, 
temperature,  rainfall,  bushels  per  acre,  dollars  per  capita,  etc. 
Given  cost  per  yard  to  find  value  of  three-eighths  or  five-eighths 
of  a  yard. 

Use  of  the  shilling. 

Use  of  fractions  in  sales-checks,  bills,  payrolls  and  piece 
labor  cards. 

Suggestions  for  Correlation 

Daily  oral  drill  in  fractions  used  in  the  occupational  work 
of  the  school,  and  in  practical  business  methods. 

Division  of  given  spaces  into  any  number  of  equal  parts  as 
needed  for  design  work. 

Finding  missing  fractional  addends  as  needed  for  putting 
dimensions  on  drawings. 

Sawing  lumber  and  cutting  cloth  into  any  number  of  equal 
parts  to  supply  each  pupil  with  one,  two  or  more  pieces. 

Original  problems  made  from  the  pupils'  mechanical  drawings. 


Ill     COMPOUND  NUMBERS 

The  Compound  Number  Tables  which  are  needed  for  buying  and 
selling  food,  clothing,  fuel,  house  furnishings,  building  materials  and 
real  estate. 

Miscellaneous  Table  as  needed  for  school  and  office  supplies. 

Table  of  Time  and  difference  in  dates. 

Reduction  of  Compound  Numbers. 

Meter  and  liter  compared  with  yard  and  quart. 

Emphasis  put  upon  the  use  of  the  yard  stick  and  tape  in  measuring 
distances  and  determining  areas  and  contents. 

Fundamental  Processes. 

Suggestions  for  Application 

Determining  cost  of  excavating,  roofing,  painting,  carpeting,  laying 
walks,  building  fences,  planting  hedges  and  gardens. 
Thermometer  readings. 
Problems  in  ventilation. 
Finding  contents  of  barrels,  cisterns,  storage  bins  and  silos. 

Suggestions  for  Correlation  (Boys) 

Emphasis  put  upon  standard  dimensions  of  lumber  as  given  on 
Price  List. 

Over-all  dimensions  in  shop  problems  estimated  from  given  lum- 
ber bills. 

Putting  price  marks  on  furniture. 

Orders  written  for  supplies  from  lumber  mills,  hardware  stores, 
etc. 

Original  problems  by  the  pupils. 

Suggestions  for  Correlation  (Girls) 
Practice  in  the  use  of  the  scales  and  measures. 
Comparison  of  weight  and  bulk  of  vegetables  bought  by  the  pound. 
Tables  used  for  recipes,  memorized. 

Multiplying  and  dividing  amounts  given  in  recipes  in  calculating 
the  cost  of  meals  for  one  or  more  persons. 

Writing  menus,  cost  being  limited. 

Determining  amount  of  money  saved  when  buying  by  the  bolt, 
piece  or  bulk,  or  by  doing  ones'  own  sewing,  cooking,  laundering 

Emphasis  put  upon  standard  widths  of  cloth. 

Finding  exact  bias  of  the  commonly  used  widths  (Square  Root). 

Over-all  dimensions  in  sewing  problems,  allowing  for  waste, 
shrinkage,  ruffles,  seams,  tucks,  hems,  etc. 

Orders  written  such  as  may  be  sent  to  clothing  and  furniture 
houses. 

Original  problems  by  the  pupils. 

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IV  UNITED  STATES  MONEY 

Kinds  of  currency  in  daily  use  in  United  States. 
Bank  Checks,  Travelers'  Checks,  Deposit  Slips,  Money  Orders 
and  Promissory  Notes. 

Suggestion  for  Application 

Pupils  will  gain  efficiency  if  allowed  to  establish  a  postal  station 
and  organize  a  bank  in  the  class  room. 

Suggestions  for  Correlation 

Household  accounts  kept  by  girls'  classes,  using  items  brought 
from  home. 

Girls  taught  to  keep  own  personal  accounts.  Neatness,  spelling 
and  correct  balancing  emphasized. 

Merchants'  bills,  electric  and  gas  bills  brought  to  class  and  com- 
pared. 

Reading  of  meters. 

Best  way  of  making  payments  of  water  rents,  taxes  and  insurance. 

Pupils  make  original  housekeepers'  problems  suggested  by  experi- 
ences at  home. 

V     PERCENTAGE  AND  ITS  APPLICATION 

The  three  types  of  problems  emphasized  and  traced  through  the 
following  subjects  using  only  practical  cases : 

Trade  Discount,  Commission,  Profit  and  Loss,  Simple  Interest, 
Taxes,  Customs  and  Duties,  and  Insurance. 

Suggestions  for  Correlation 

Commission  and  Profit  and  Loss  emphasized  in  selling  domestic 
science  and  manual  training  products. 

VI     RATIO 

Compare  terms  with  terms  of  a  fraction. 

Suggestion  for  Application 

Use  Building  Code  for  original  problems  such  as  finding  ratio  of 
light  surface  to'  floor  surface,  rise  of  stairs  to  tread,  pitch  of  roofs  to 
wind  pressure,  etc. 

VII     PROPORTION  AND  THE  EQUATION 

The  four  axioms  involving  the  four  fundamental  processes. 
Show  which  signs  are  indicated  by  finding  numerical  value. 

Suggestions  for  Application 

Mixing  of  liquids,  grains,  fertilizers,  etc. 

Heights  of  building  found  by  comparison  of  shadows. 

Relative  speeds  of  pulleys,  shafts,  and  machines.  , 

Supply  and  demand. 

Raw  material  and  finished  products. 

Proportion  in  Partnership. 

Suggestion  for  Correlation 

The  equation  is  shown  on  a  drawing  for  cabinet  work;  the  height 
of  a  chair  is  equal  to  a  total  of  the  heights  of  the  parts,  and  again 
these  parts  are  divided  to  show  joints  and  ornamentation. 


Find  missing  fractional  dimensions  when  totals  are  given. 
Pupils  test  mill  bills  on  their  blue  prints  from  drawing  room. 
From  measurements  taken  in  the  shop  estimate  speeds  of  machines. 

VIII     MENSURATION  AND  CONSTRUCTION  PROBLEMS 

Erecting  perpendiculars  and  drawing  parallel  lines. 

Dividing  lines  into  any  number  of  equal  parts. 

Dividing  of  the  circle  into  degrees. 

Measuring  angles. 

Bisecting  angles. 

Constructing  squares,  rectangles,  trapezoids,  equilateral  and  right 
triangles,  hexagons  and  circles.  Find  areas  of  same. 

Compare  perimeters  of  figures  having  equivalent  areas. 

Ratio  of  the  perimeters  of  parallelograms  having  equivalent  areas. 

Inscribing  square  and  hexagon  within  a  circle  and  drawing  tan- 
gents. 

Surface  of  cylinder,  cone  and  square  pyramid. 
Suggestions  for  Application 

Finding  areas  of  the  four  figures  usually  found  in  roofs,  and 
exposed  surfaces  of  cone-shaped  or  pyramid-shaped  towers. 

Practical  use  of  the  triangle  in  estimating  heights  and  distances 
impossible  to  measure. 

Suggestions  for  Correlation 

Relate  geometrical  construction  to  shop  problems,  art  work, 
making  geographical  charts,  etc. 

Models  made  in  the  shop  to  illustrate  formulas. 

Drawings  and  hand  printed  charts  made  to  emphasize  rules  and 
principles. 

Account  book  covers  made  in  the  art  room. 

GENERAL  SUGGESTIONS 

(a)  When  reading  numbers  use  the  word  "and"  to  indicate  the  deci- 

mal point  only. 

(b)  When  possible  indicate  division  in  the  form  of  a  fraction  in 

the  planning  of  problems. 

(c)  Have  pupils  estimate  results  before  solving. 

(d)  Occasionally  have  the  steps  of  problems  indicated  without  solv- 

ing. 

(e)  Note  that  the  quotient  obtained  in  ratio  is  always  abstract. 

(f)  When  classes  take  their  trips  to  municipal  buildings,  banks,  fac- 

tories and  stores,  have  them  use  the  information  received  in 
writing  original  problems.  Such  problems  should  be  tested, 
solved  and  collected  by  the  classes. 

(g)  Insist  upon  correct  spelling  and  punctuation  in  all  original  prob- 

lems, accounts  and  recipes, 
(h)     Give    pupils    approbation    for    questions    which    show    careful 

thought, 
(i)      Help  the  pupil  to  see  that  what  he  is  required  to  do  is  practical. 

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